Upload
others
View
14
Download
0
Embed Size (px)
Citation preview
Bus Arrival Time Estimation Using Traffic Signal Database & Shockwave Theory Md. Matiur Rahman
1, Lina Kattan
2, S.C. Wirasinghe
3 and Doug Morgan
4
12th WCTR, July 11-15, 2010 – Lisbon, Portugal
1
BUS ARRIVAL TIME ESTIMATION USING TRAFFIC SIGNAL DATABASE &
SHOCKWAVE THEORY
1. Md. Matiur Rahman, Graduate Student, Department of Civil Engineering, Schulich School of Engineering, University of Calgary, Canada.E-mail:[email protected]
2. Lina Kattan, Assistant Professor, Department of Civil Engineering, Schulich School of Engineering, University of Calgary, Canada. E-mail: [email protected]
3. S.C. Wirasinghe, Professor and P.I. PUTRUM Research Program, Department of Civil Engineering, Schulich School of Engineering, University of Calgary, Canada. E-mail:[email protected]
4. Doug Morgan, Manager, Transit Operations, Calgary Transit, Calgary, Canada.
ABSTRACT
Intersection delays are one of the major sources of uncertainty in real-time bus arrival time
estimation. Only a few studies have included signal delay in estimating bus arrival time. In
this paper, a new approach is developed to incorporate intersection delays in bus arrival time
estimation. The approach integrates information from: 1) an existing signal database, 2)
estimated speed of shockwaves created by a red signal and 3) bus speed available from
GPS tracking. From the online access to a signal timing plan, the exact signal phasing of any
fixed signalized intersection is known in real time. Hence, the signal phase and elapsed time
of that phase can be determined. In addition, the developed framework assumes that traffic
conditions as well as bus speed are known or can be determined. Estimated bus travel time
between a bus stop and a signalized intersection is divided into two parts: 1) cruising travel
time and 2) travel time in queue. Cruising travel time is defined as the bus travel time
required to join the backward propagating shockwave from the traffic light. Travel time in
queue is obtained from the estimated delay in the queue. Two illustrative numerical
examples show that this approach can incorporate the intersection delay well in real-time bus
arrival time estimation.
Keywords: bus arrival time, existing signal database, shockwaves
Bus Arrival Time Estimation Using Traffic Signal Database & Shockwave Theory Md. Matiur Rahman
1, Lina Kattan
2, S.C. Wirasinghe
3 and Doug Morgan
4
12th WCTR, July 11-15, 2010 – Lisbon, Portugal
2
1. INTRODUCTION
Delays at signalized intersections are important factor to consider in estimating real time bus
arrival time at a given bus stop location. A number of models have been developed so far to
provide real time bus arrival information [1,2,4,6,10,13-15,18] based on real-time bus GPS
tracking systems. In these models, delays due to a signalized intersection are incorporated
indirectly. None of existing models consider intersection delays explicitly and the effect of
queue formation due to a red signal. During the red signal, a backward (primary) shockwave
which travels opposite to the intersection approach is formed and defines the rear end of the
queue. During the next green phase, another backward shockwave, known as the recovery
shockwave forms as well. The speed of the recovery shockwave is greater than the primary
shockwave and after some time it catches the primary shockwave and restores the normal
traffic condition. Based on online monitoring of prevailing traffic condition (i.e. flow, density)
speed of a shockwave can be determined by applying classical shockwave theory to a pre-
timed signal [7, 9]. When the bus departs from the bus stop, its real time speed can be
estimated by integrating information from: 1) real-time GPS tracking and 2) the historical bus
speed data. After obtaining bus speed and shockwave speed, the link bus travel time can be
estimated by considering various traffic conditions that the bus might face.
2. LITERATURE REVIEW
Detailed reviews of different approaches to bus travel time estimation are discussed by
Chien et al [2]. Most available models can be classified into the following groups:
Regression [13]
Analytical [8,10,15,16]
Kalman Filters [4,6,14,18]
Artificial Neural Networks (ANN) [2,6]
Combination of ANN & Kalman Filters [1,12]
Regression models (both linear and nonlinear) are well suited for parameter estimation
problems due to their simplicity and ease of use [13]. Kalman filtering approaches that have
elegant mathematical representations (e.g. linear state-space equations) have been applied
in the estimation of bus travel times. The Kalman filter approaches have potentials to
incorporate traffic fluctuations with their time-dependent parameters (e.g. Kalman gain)[2].
Artificial Neural Networks (ANN) models can map the non linear relation among the
dependent and independent variables.. Son et. al. [14] proposed the segmentation of link
between bus stops at the signalized intersections; still the authors did not considered the
effect of real time queue estimation. The combinations of ANN and Kalman Filter approaches
have shown promising results in improving the estimation of bus travel time [1, 12]. However,
none of these approaches have considered the delays due to a signalized intersection
explicitly [4, 6, 18 2, 6]. In this paper we propose a method to estimate the bus delay at a
signalized intersection, in real time. This delay can be incorporated with the existing bus
travel time estimation methods to improve accuracy.
Bus Arrival Time Estimation Using Traffic Signal Database & Shockwave Theory Md. Matiur Rahman
1, Lina Kattan
2, S.C. Wirasinghe
3 and Doug Morgan
4
12th WCTR, July 11-15, 2010 – Lisbon, Portugal
3
3. PROBLEM STATEMENT
Intersection delays are one of the major sources of uncertainty in real-time bus arrival time
estimation. In this paper, a new approach is developed to incorporate intersection delays in
bus arrival time estimation. During the red signal a backward shockwave is formed. This
shockwave travels backward and defines the rear end of queue. When the traffic signal turns
green a recovery shockwave is formed and travels backward too. The speed of recovery
shockwave is greater than the backward shockwave and after some time it catches the
primary backward shockwave and restores the normal traffic condition. When the bus
departs from the bus stop, its speed can be estimated based on the archived speed data
from historical GPS tracking. Details of the traffic signal timing plan are known for pre-timed
signals and speed of shockwaves can be estimated based on prevailing traffic condition (i.e.
flow, density). This study focuses solely on a pre-timed signal. This paper proposes a
framework to integrate information from: 1) an existing signal database, 2) estimated speed
of shockwaves created by a red signal and 3) bus speed available from GPS tracking. The
major contribution of this study is to incorporate the delays created by traffic signal operation
along the bus routes in bus travel time estimation.
4. PROPOSED APPROACH
The various possible bus trajectories for a pre-timed signalized intersection are illustrated in
the Figures 1 and 2. Figure 1 illustrates the case of moderate congestion (i.e. no residual
queue from previous cycle) and Figure 2 illustrates the case of oversaturated conditions
where the traffic signal was not able to dissipate the queue completely.
Figure 1: Time-space diagram without residual queue
Bus Arrival Time Estimation Using Traffic Signal Database & Shockwave Theory Md. Matiur Rahman
1, Lina Kattan
2, S.C. Wirasinghe
3 and Doug Morgan
4
12th WCTR, July 11-15, 2010 – Lisbon, Portugal
4
As can be seen from Figure 1 the bus can follow different travel trajectories between bus
stop k and stop line k+1. This bus trajectory depends on the departure time of the bus from
the bus stop k as well as prevailing traffic conditions. Without any residual queue (the queue
from the previous cycle is fully discharged in the last green phase), the bus can possibly
arrive at the traffic signal k+1 during the red time period, so it will have to wait (Trajectory
Type1 in Figure 1). Another possible trajectory represents the bus arrival at the signal just
when the signal turns green. In that latter case, the bus may experience some delays due to
the time needed to discharge the queue in front of the bus (Trajectory Type 2). Another
possible trajectory represents the bus passing the traffic signal without any delay (Trajectory
Type 3).
Figure 2: Time-space diagram with residual queue
Figure 2 illustrates the possible trajectories with the presence of a residual queue from a
previous cycle. It shows another possible bus trajectory where the bus joins the intersection
queue during the green phase but faces the back propagating shockwave and hence is not
able to cross the intersection within that green phase (Trajectory Type 4).
The above possible trajectories show that the bus arrival times at downstream bus stop k+2
are not similar and vary significantly depending on the prevailing traffic conditions and signal
condition at the traffic signal k+1. By tracking busses with GPS, the real-time bus departure
time from a bus stop k can be obtained. Prevailing traffic conditions at the intersection at a
given time interval can be estimated from historical data. In addition, the exact signal phasing
for pre-timed signals are also available and can be used to obtain the exact signal condition
corresponding to a given bus departure time from a bus stop. These different sources of
information can be integrated to estimate the bus trajectories thus bus travel time. Cruising
bus speeds as well as speed of shockwaves are estimated for the link k after evaluating the
state of downstream signalized intersection. The estimation process can be divided into main
4 steps:
Bus Arrival Time Estimation Using Traffic Signal Database & Shockwave Theory Md. Matiur Rahman
1, Lina Kattan
2, S.C. Wirasinghe
3 and Doug Morgan
4
12th WCTR, July 11-15, 2010 – Lisbon, Portugal
5
Step 1: Estimation of bus speed
Since the length of the road link is known, mean bus cruising speed can be estimated using
the real-time as well as historical GPS speed track database.
Step 2: Evaluation of traffic signal
The exact signal phasing of pre-timed traffic signal at the downstream intersection can
obtained on real-time. So the signal state corresponding to the bus departure time can be
determined.
Step 3: Estimation of shockwave speed
Prevailing traffic condition (i.e. flow, density, speed) is obtained in real-time from advanced
detector data. Thus, the speed of shockwaves (i.e. backward or forward) can be estimated
from classical shockwave theory.
Step 4: Estimation of bus link travel time
After getting the estimated bus and shockwave speeds, link travel time of bus can be
estimated as explained in Section 6.
5. DETERMINATION OF SHOCKWAVE SPEED
Traffic condition is mainly defined by the following three fundamental traffic parameters:
space speed, density and flow. When these parameters change suddenly such as in the
case of a red light or sudden increase in density, an edge or boundary is established that
makes a distinction between two flow conditions. This boundary is called a shockwave in the
context of traffic flow theory [3]. Wirasinghe [19] showed how individual and overall traffic
delays at various types of bottlenecks could be estimated using shockwave analysis. The
behavior of traffic shockwaves was demonstrated by Lighthill and Whitham [9]. Liu et al [11]
estimated queue lengths by tracing the trajectory of shockwaves based on the continuum
traffic flow theory. Traffic shockwave can be described by using flow-density (q-k) diagram.
The backward shockwave, V2 and the recovery shockwave, V3 are shown in the Figure 3 and
are explained next.
Figure 3: Representation of shockwaves in the fundamental diagram.
Bus Arrival Time Estimation Using Traffic Signal Database & Shockwave Theory Md. Matiur Rahman
1, Lina Kattan
2, S.C. Wirasinghe
3 and Doug Morgan
4
12th WCTR, July 11-15, 2010 – Lisbon, Portugal
6
Let us denote, q1, k1, u1 are the flow, density and velocity of the upstream region and q2, k2, u2
are the flow, density and velocity of the downstream region. From Lighthill-Whitham-Richards
(LWR) traffic flow model, the shockwave velocity can be written as below:
12
12
kk
k
qV
Garber, N.J. and Hoel, L.A. (7) suggested a more simplified formula to estimate shockwave
speed for the traffic approaching a red light. Backward shockwave speed is estimated as: V2
= Vf η1 and recovery shockwave is estimated as: V3 = Vf, where Vf is free flow speed and η1 is
the ratio of upstream density and jam density. Thus, when flows and densities are known,
shockwave velocities can be determined.
6. MODEL DESCRIPTION
Travel time between a bus stop and a signalized intersection is divided into two parts:
Cruising travel time of bus, Tc,k which is defined as the travel time of a bus to reach
the queue, i.e. time required to travel the path (Lk – X) as indicated in Figure 4,
Travel time in queue, Td,k which is defined as the estimated travel time in the queue
i.e. time required to travel the distance X as indicated in Figure 4
Figure 4: Layout of hypothetical road link.
Estimation of bus link travel time for link, k+1 can be directly determined from the estimated
speed of the bus on this link where the link travel time is the link length divided by the bus
speed. The detailed procedure for determination of Tc,k and Td,k for the link, k are discussed
below. In what follows, bus speed is denoted as,V1,i , backward shockwave speed due to red
signal is denoted as V2,i and recovery shockwave speed due to green signal is denoted as
V3,i. For a moderate congestion condition i.e. no residual queue at a signalized intersection,
these speeds are assumed to be known. In this analysis, green signal time includes the
green and yellow phases and the aforementioned three speeds are assumed to be constant
for simplicity. When a bus departs from a bus stop, the signal of the downstream intersection
may be either red or green. If the signal is red, there will be a moving backward shockwave
Bus Arrival Time Estimation Using Traffic Signal Database & Shockwave Theory Md. Matiur Rahman
1, Lina Kattan
2, S.C. Wirasinghe
3 and Doug Morgan
4
12th WCTR, July 11-15, 2010 – Lisbon, Portugal
7
towards the bus and if the bus stop is close enough, the bus can meet this shockwave within
this red phase but if bus stop is far away from the intersection, the bus may take more than
one cycle to cross the intersection. The schematic diagram of estimation of link travel time is
shown in the following figure:
Figure 5: Schematic diagram of bus link travel time estimation
As can be seen in Figure 5, in this approach, the 2 possible cases of signal state are
examined:
Case 1: downstream signal is red when the bus leaves the upstream bus stop and
Case 2: downstream signal is green when the bus leaves the upstream bus stop
As mentioned previously, the required travel time for the bus to cross the intersection is
computed based on: estimated bus speed, 2) estimated shockwave speed and 3) prevailing
signal conditions.
Case 1: Traffic Signal is Red at Downstream Intersection k+1 when the Bus Departs from Bus Stop k.
Duration of Red phase = Ri , Total Green time Duration = Gi, Elapsed time (i.e. time from
start of red) = Ei, Remaining time for the signal to turn green= (Ri-Ei), Distance between bus
stop and stop line = Lk (Figure 4). For both cases, the number of phases or cycles required
for the bus to reach the downstream intersection is first determined then the link travel time
of bus is estimated. This will be achieved by applying the following steps:
Step 1:
This step checks whether the bus will be able to arrive at the intersection (k+1) within the
current red phase (phase i). This will mainly depends on: 1) the link length, 2) bus speed, 3)
shockwave speed and 4) duration of red phase. As indicated in Figure 6 below, the queue
Evaluation of bus departure time at bus stop, k
Evaluation of signal state of signalized intersection, k+1
Case 1: Red Case 2: Green
Estimation of cruising bus speed and shockwaves speed
Determination of number of phases or cycles required to reach around the intersection for bus
Estimation of cruising travel time Estimation of travel time in queue
Estimation of link travel time
Bus Arrival Time Estimation Using Traffic Signal Database & Shockwave Theory Md. Matiur Rahman
1, Lina Kattan
2, S.C. Wirasinghe
3 and Doug Morgan
4
12th WCTR, July 11-15, 2010 – Lisbon, Portugal
8
length created by backward shockwave V2,i in elapsed time, Ei is V2i *Ei. Due to the
propagating backward shockwave, this queue will be growing till the signal turn greens.
Thus, the additional queue length till the signal turns green is V2i *(Ri-Ei). The bus is travelling
at a speed of V1,i when leaving the upstream bus stop. The distance travelled by bus in time
interval (Ri-Ei) is V1i*(Ri-Ei). If the summation of these distances is equal or greater than the
link distance, Lk then the bus will join the queue within this phase; otherwise this bus will not
be able to cross the intersection within this red phase. This criterion can be written
mathematically as below:
Checking whether, V2i *Ei + V2i *(Ri-Ei) + V1i*(Ri-Ei) ≥ Lk ----- (1)
If the summation of the left side of Eqn.1 is greater than the right side, the bus will join the
queue within the current red phase; otherwise we have to check again for the next green
phase as illustrated in step 2. Let’s denote ti, as the time needed for the bus to join the
intersection queue as indicated in Figure 6:
Figure 6: Schematic diagram of road link, Lk
If the bus meets the back propagating shockwave during the first red phase, implying that the
summation of the three distances indicated in Figure 6 is equal to link distance, Lk. this can
be written mathematically as follows:
V2,i *Ei + V2,i *ti + V1,i*ti = Lk ----- (2)
=> ii
iiki
VV
EVLt
,2.1
,2 *
Since during time period ti, the bus will not face any intersection queue, Cruising travel
time, Tc,k is equal to ti. The total queue length in front of bus (i.e. red shaded area in Figure
6) is estimated as V2,i *Ei + V2,i *ti. If average spacing of vehicles is Sv, then number of
vehicles, Nk,i in front of bus can be estimated as below:
v
iiiiik
S
tVEVN
** ,2,2,
After joining the queue at the downstream intersection, the bus will remain stopped till the
start of next green phase. The bus waiting time within red signal time is computed as:[Ri –
(Ei+ti)]. When the traffic signal turns green, the queue in front of bus will start to dissipate at
saturation flow rate. The time required to dissipate the queue in front of bus is [Nk,i/ qs (t)],
where qs (t) is saturation flow rate. Bus travel time in the queue is computed as:
)()]([
,,
tq
NtERT
s
ikiiikd
V1,i*ti V2,i *ti V2,i *Ei
Intersection: k+1 Bus stop: k Link, Lk
Queue length
during Ei
Queue being
formed during ti
Distance being travelled
by bus during ti
Bus Arrival Time Estimation Using Traffic Signal Database & Shockwave Theory Md. Matiur Rahman
1, Lina Kattan
2, S.C. Wirasinghe
3 and Doug Morgan
4
12th WCTR, July 11-15, 2010 – Lisbon, Portugal
9
Step 2:
If V2i *Ei + V2i *(Ri-Ei) + V1i*(Ri-Ei) < Lk, the bus will not be able to arrive at the intersection
(k+1) within the first red phase (phase i) as indicated in Figure 7, below:
Figure 7: Schematic diagram of road link, Lk
Thus, in this case, the bus will not join the back of the queue during the current red phase.
However, depending on the link length, bus speed, shockwave speed and duration of the
next green phase, the bus might still be able to cross the (k+1) within this green phase
(phase i).
If time required for the queue formed in the last red phase to be dissipated is di , V2,i(G) is the
backward shcokwave speed during this green phase and V3,i is the recovery shockwave
speed due to green signal, di can be computed as:
V3i* di = V2i *Ri + V2i(G) * di
=> di = (V2i*Ri )/( V3i - V2i (G))
During the time di, there will be some additional queue forming until the recovery backward
shockwave catches the previous primary backward shockwave. Thus, the bus might stil be
able to catch the back of the queue during di,this might happen if:
V1,i *(Ri –Ei) + V1,i(G) * di + V2,i *Ri + V2,i(G)*di –V3,i *di ≥ Lk --------(3).
Where, V1,i(G) is bus speed during this green phase. If Eqn.3 is satisfied, then the link diagram
can be presented as illustrated in Figure 8 below:
Figure 8: Schematic diagram of road link, Lk
At the start of the green phase (i.e. during di), there will be additional queue forming until the
recovery backward shockwave catches the previous primary backward shockwave. If the bus
comes at intersection in this duration (i.e. during di), it will face the queue and the time
required,ti for the bus to meet the queue can be written as below:
V1,i *(Ri –Ei) + V1,i(G) * ti + V2,i *Ri + V2,i(G)*ti –V3,i *ti = Lk --------- (4)
=> iGiGi
iiiiiki
VVV
RVERVLt
,3)(,2)(.1
,2,1 *)(*
So, Cruising travel time: Tc,k = (Ri –Ei) + ti and Travel time in queue can be estimated as
)(
,,
tq
NT
s
ikkd
, v
iiiGiiiik
S
tVtVRVN
*** ,3)(,2,2,
V1,i(G) * ti
Distance being travelled by
bus in ti
V1,i *(Ri –Ei) V2,i(G)*ti V2,i *Ri- V3,i *ti
Intersection: k+1 Bus stop: k Link, Lk
Queue being
formed during ti
Distance travelled by bus in
last red phase
V1,i*(Ri-Ei) V2,i *(Ri-Ei) V2,i *Ei
Intersection: k+1 Bus stop: k Link, Lk
Queue length
during Ei
Queue formed
during (Ri-Ei)
Distance travelled by bus
during (Ri-Ei)
Bus Arrival Time Estimation Using Traffic Signal Database & Shockwave Theory Md. Matiur Rahman
1, Lina Kattan
2, S.C. Wirasinghe
3 and Doug Morgan
4
12th WCTR, July 11-15, 2010 – Lisbon, Portugal
10
If equation 3 is not satisfied, we have to go to step 3 as illustrated in what follows.
Step 3:
In this case, V1,i *(Ri –Ei) + V1,i(G) * di + V2,i *Ri + V2,i(G)*di –V3,i *di < Lk. Thus, the bus will not
join the back of the queue during this green phase. However, depending on the link length,
bus speed, shockwave speed and duration of the green phase, the bus might still be able to
cross the intersection during the green phase. This will happen if Eqn. 5 below is satisfied:
V1,i *(Ri –Ei)+ V1,i(G) * Gi > Lk --------(5)
If V1,i *(Ri –Ei)+ V1,i(G) * Gi > Lk, then bus will be able to cross the intersection within this
green phase without facing any intersection delay, the schematic link diagram will look like as
Figure 9:
Figure 9: Schematic diagram of road link, Lk
As shown in Figure 9, in such situation (i.e. V1,i *(Ri –Ei)+ V1,i(G) * Gi > Lk) , bus will not face
intersection delay. So, V1i *(Ri –Ei) + V1,i(G) * ti = Lk, => )(.1
,1 )(*
Gi
iiiki
V
ERVLt
Bus Cruising Travel time: Tc = (Ri –Ei) + ti. In this case, there is no waiting time for bus in this
intersection. So travel time in queue: Td,k = 0 s
If equation 5 is not satisfied then, the model will proceed to the next step 4.
Step 4:
If V1,i *(Ri –Ei)+ V1,i(G) * Gi < Lk, bus will not be able to cross the intersection within this green
phase (i.e. phase i). Similar to step 1, Check for the next red phase whether V1,i *(Ri –Ei)+
V1,i(G) * Gi + V1,i+1*Ri+1 +V2,i+1*Ri+1 ≥ Lk. If yes, then use V1,i *(Ri –Ei)+ V1,i(G) * Gi + V1,i+1* ti+1
+V2,i+1* ti+1 = Lk
=> 1,21,1
)(,1,11
*)(*
ii
iGiiiiki
VV
GVERVLt
Cruising Travel time: Tc,k = (Ri –Ei) +Gi + ti+1
Travel time in queue:)(
)(1,
11,tq
NtRT
s
ikiikd
Where,
v
iiik
S
tVN
11,21,
*
Step 5:
If V1,i *(Ri –Ei)+ V1,i(G) * Gi + V1,i+1*Ri+1 +V2,i+1*Ri+1 < Lk. , then bus will not be able to cross the
intersection within this red phase (phase i+1). As discussed in step 2, if total Green time =
V1,i(G) * ti
Distance being travelled by
bus in ti
V1,i *(Ri –Ei)
Intersection: k+1 Bus stop: k Link, Lk
Distance travelled by bus in
last red phase
Bus Arrival Time Estimation Using Traffic Signal Database & Shockwave Theory Md. Matiur Rahman
1, Lina Kattan
2, S.C. Wirasinghe
3 and Doug Morgan
4
12th WCTR, July 11-15, 2010 – Lisbon, Portugal
11
Gi+1 and time required to be dissipated the queue formed in the last red phase is di+1 then
V3,i+1* di+1 = V2,i+1 *Ri+1 + V2,i+1(G) * di+1, => di+1 = (V2,i+1*Ri+1 )/( V3,i+1 - V2,i +1(G)) and if V1,i *(Ri –
Ei)+ V1,i(G) * Gi + V1,i+1*Ri+1 + V1,i+1(G)*di+1 +V2,i+1*Ri+1 + V2,i+1(G)*di+1-V3,i+1*di+1 > Lk then V1,i *(Ri –
Ei)+ V1,i(G) * Gi + V1,i+1*Ri+1 + V1,i+1(G)*ti+1 +V2,i+1*Ri+1 + V2,i+1(G)*ti+1 - V3,i+1*ti+1 = Lk , so
1,3)(1,2)(1.1
11,211,1)(,1,11
***)(*
iGiGi
iiiiiGiiiiki
VVV
RVRVGVERVLt
Cruising travel time: Tc,k = (Ri –Ei) +Gi + Ri+1 + ti+1 and Travel time in queue:)(
1,,
tq
NT
s
ikkd
Where,v
iiiGiiiik
S
tVtVRVN
11,31)(1,211,21,
***
Step 6:
If V1,i *(Ri –Ei)+ V1,i(G) * Gi + V1,i+1*Ri+1 + V1,i+1(G)*di+1 +V2,i+1*Ri+1 + V2,i+1(G)*di+1-V3,i+1*di+1 <Lk, bus
will not face the queue in this green phase at least. Then it will be checked whether V1,i *(Ri –
Ei) + V1,i(G) * Gi + V1,i+1*Ri+1 +V2,i+1*Ri+1 + V1,i+1(G)*Gi+1 > Lk. If yes, then V1,i *(Ri –Ei) + V1,i(G) * Gi +
V1,i+1*Ri+1 +V2,i+1*Ri+1 + V1,i+1(G)*ti+1 = Lk
=> )(1.1
11,211,1)(,1,11
***)(*
Gi
iiiiiGiiiiki
V
RVRVGVERVLt
Cruising Travel time: Tc,k = (Ri –Ei) +Gi + Ri+1 + ti+1 And there will be no waiting time for bus.
So travel time in queue: Td,k = 0 s
Step 7:
If V1,i *(Ri –Ei)+ V1,i(G) * Gi + V1,i+1*Ri+1 +V2,i+1*Ri+1 + V1,i+1(G)*Gi+1 < Lk, bus will not be able to
cross the intersection within this green phase. And the above procedure will be repeated for
the next signal phases until it will get the solution.
Case 2: Traffic Signal is Green at Downstream Intersection k+1 when the Bus Departs from Bus Stop k.
Total Green time = Gi, Elapsed time = Ei, Remaining time = (Gi-Ei). As discussed for Case 1,
Step 2, if the time required to dissipate the queue formed in the last red phase is di then
V3,i*di = V2,i *Ri + V2,i(G) *di => di = (V2,i *Ri )/( V3,i - V2,i(G) )
If the distance travelled by the bus during the remaining time of the green phase is greater
than Lk, and if the Bus is not joining the back of the queue, the bus will not be delayed due to
the queue and will only be travelling at the cruising speed. This situation will occur if V1i *(Gi
–Ei) > Lk and Ei > di. Thus, the bus travel time on link Lk can be determined as ti = Lk / V1i.
Thus, Cruising Travel time: T c,k = ti and waiting time in queue will be zero, so travel time
Bus Arrival Time Estimation Using Traffic Signal Database & Shockwave Theory Md. Matiur Rahman
1, Lina Kattan
2, S.C. Wirasinghe
3 and Doug Morgan
4
12th WCTR, July 11-15, 2010 – Lisbon, Portugal
12
in queue: Td,k = 0 s. However, if the bus joins the back of the queue (i.e. if Ei < di) during this
green time, then the bus will be travelling at 2 different speeds: cruising speed and speed in
the queue. The time required to catch the primary backward shockwave by recovery
shockwave is denoted as (di-Ei) and time required to meet the bus and the 1st backward
shockwave is ti. Thus, the bus will join the back of the queue if the following equation holds:
V1,i*(di - Ei) + V2,i *Ri +V2,i(G) *Ei + V2,i (G)*(di - Ei) – V3,i*(di - Ei) > Lk --------(6)
The cruising travel time of bus, T c,k = ti is estimated as follows:
V1,i*ti + V2,i *Ri +V2,i(G) *Ei + V2,i (G)*ti - V3,i* ti = Lk => iGii
iGiiiki
VVV
EVRVLt
,3)(,2,1
)(,2,2
-
**
And travel time in queue:)(
,,
tq
NT
s
ikkd Where,
v
iiiGiiiik
S
tVtVRVN
*** ,3)(,2,2,
If Eqn.6 does not hold i.e. V1,i*(di - Ei) + V2,i *Ri +V2,i(G) *Ei + V2,i (G)*(di - Ei) < Lk, the bus will not
face the queue in this green phase and then V1i *ti = Lk => ti = Lk / V1i. Cruising Travel time:
Tc,k = ti = Lk / V1i and waiting time in queue will be zero. So travel time in queue: Td,k = 0 s. If
V1i *(Gi –Ei) < Lk then bus will not be able to cross the intersection within this green phase
and the model will proceed to next red phase and the aforementioned steps of Case 1 will be
applied.
For each green phase, the length of residual queue can be estimated by deducting the total
discharged vehicles during a given green phase from the total vehicles arrived in the last red
phase. For the case of presence of residual queue, the main difference will be in checking
the effective link length. Instead of Lk, the check will be applied to (Lk - QR,i-1), where QR,i-1 is
the residual queue. The queue due to next red signal will form behind this residual queue. As
for example, if signal phase is red when the bus leaves the upstream bus stop, condition: V2i
*Ei + V2i *(Ri-Ei) + V1i*(Ri-Ei) > (Lk - QR,i-1) is checked and the procedure will be same as the
previously described that determines cruising travel time. In estimation of travel time in
queue, another additional check has to be performed, whether the bus will be able to pass
the intersection with the remaining available green time of the current phase. If time required
to discharge of the queue in front of bus is more than the green time, then the bus will not be
able to pass the intersection in this green phase as well as in the next red phase which can
be estimated by simple modification in the aforementioned equations.
7. MODEL ILLUSTRATION BY NUMERICAL EXAMPLES
For simplicity, it is assumed that every signal cycle starts with its red phase in a two phase
signalized intersection. If a bus departs from bus stop, k at ith cycle of downstream
intersection, let us denote the following:
Cycle Length, Ci = 30 sec, Red Duration, Ri = 15 sec, Green Duration, Gi = 15 sec
Here, two numerical examples are shown to represent two important scenarios. Example 7.1
corresponds to short distance between bus stop and downstream signal and Example 7.2
represents the case where the distance between bus stop and downstream signal is long
Bus Arrival Time Estimation Using Traffic Signal Database & Shockwave Theory Md. Matiur Rahman
1, Lina Kattan
2, S.C. Wirasinghe
3 and Doug Morgan
4
12th WCTR, July 11-15, 2010 – Lisbon, Portugal
13
and more than one signal cycles may be required to cross the intersection, k+1 from bus
stop, k for bus.
Example 7.1
Link length (distance between bus stop & stop line), Lk = 70 meter
Departure time, Dk = 9.00.000
Step 1: Evaluate the departure time of bus and traffic signal condition of next intersection at
that time.
Let’s assume the signal of downstream intersection, k+1 is red when bus departs from bus
stop, k.
Elapsed time, Ei = 5 sec
Remaining time = (Ri - Ei) = (15-5) sec = 10 sec
Step 2: Estimation of required information (i.e. using Kalman Filter, ANN or other available
methods):
Bus speed, V1,i = 18 km/hr (known from GPS tracking database) = 18*0.278 m/s = 5.004 m/s
1st backward shockwave speed due to red signal, V2,i =8 km/hr =8*0.278 m/s = 2.224 m/s
Step 3: Determination of number of phases or cycles required to reach around the
intersection for bus
(1) Check for this red phase, whether V2i *Ei + V2i *(Ri-Ei) + V1i*(Ri-Ei) >Lk
2.224*5 + 2.224*10 + 5.004 *10 = 83.4 m > 70 m, so bus will meet the queue within this red
phase. So if time required meeting the bus and 1st backward shockwave is ti then it can be
written as: V1,i*ti + V2,i *Ei + V2,i *ti = Lk
sec146.8224.2004.5
5*224.270*
,2.1
,2
ii
iiki
VV
EVLt
Step 4: Determination of link travel time
Cruising Travel time: Tc,k = ti = 8.146 sec
Travel time in queue:
87.46
146.8*224.25*224.2** ,2,2,
v
iiiiik
S
tVEVN
Link travel time, Tk = Tc,k + Td,k =8.146+12.922 = 21.068 sec
Example 7.2:
Link length (distance between bus stop & stop line), Lk = 250 meter
Step 1: Evaluate the departure time of bus and traffic signal condition of next intersection at
that time.
Let’s assume the signal of downstream intersection, k+1 is red when bus departs from bus
stop, k and elapsed time, Ei = 5 sec and remaining time = (Ri - Ei) = (15-5) sec = 10 sec
stq
NtERT
s
ikiiikd 922.12
44.0
87.4)146.8515(
)()(
,,
Bus Arrival Time Estimation Using Traffic Signal Database & Shockwave Theory Md. Matiur Rahman
1, Lina Kattan
2, S.C. Wirasinghe
3 and Doug Morgan
4
12th WCTR, July 11-15, 2010 – Lisbon, Portugal
14
Step 2: Estimation of required information (i.e. using Kalman Filter, ANN or other available
methods):
Bus speed in red phase, V1,i = 18 km/hr = 18*0.278 m/s = 5.004 m/s
Bus speed in next green phase, V1,i(G) = 19 km/hr = 19*0.278 m/s = 5.282 m/s
1st backward shockwave speed due to red signal, V2,i =8 km/hr =8*0.278 m/s = 2.224 m/s
1st backward shockwave speed during the green phase, V2,i(G) =8 km/hr =8*0.278 m/s = 2.224
m/s
Recovery shockwave speed due to green signal, V3,i = 20 km/hr = 20*0.278 m/s = 5.56 m/s
Step 3: Determination of number of phases or cycles required to reach around the
intersection for bus
(1) Check for this red phase, whether V2i *Ei + V2i *(Ri-Ei) + V1i*(Ri-Ei) >Lk
2.224*5 + 2.224*10 + 5.004 *10 = 83.4 m < 250 m, so bus will not be able to reach around
the intersection within this red phase.
(2) Check for the next green phase
(i) Time required to be dissipated the queue formed in the last red phase = di
V3i* di = V2i *Ri + V2,i(G) * di
sec000.10224.256.5
15*224.2*
)(,2.3
,2
Gii
iii
VV
RVd
(A) Check whether V1,i *(Ri –Ei) + V1,i(G) * di + V2,i *Ri + V2,i(G)*di - V3i* di > Lk
=> 5.004*10 + 5.282*10 + 2.224*15 + 2.224*10 – 5.56*10= 102.86 m < 250m, so bus will not
face the queue in this green phase at least.
(a) Check whether V1,i *(Ri –Ei)+ V1,i(G) * Gi > Lk
=> 5.004*10 + 5.282*15 = 129.27 m < 250 m, bus will not be able to cross the intersection
within this green phase.
(3) Check for the next red phase whether V1,i *(Ri –Ei)+ V1,i(G) * Gi + V1,i+1*Ri+1 +V2,i+1*Ri+1 >
Lk
=> 5.004*10 + 5.282*15 + 5.282*15 +2.502*15 = 241.86 m < 250 m, bus will not meet queue
within this red phase
(4) Check for the next green phase
(i) Time required to be dissipated the queue formed in the last red phase = di+1
V3,i+1* di+1 = V2,i+1 *Ri+1 + V2,i+1(G) * di+1
sec385.10502.2116.6
15*502.2*
)(1,21,3
11,21
Gii
iii
VV
RVd
(A) Check whether V1,i *(Ri –Ei)+ V1,i(G) * Gi + V1,i+1*Ri+1 + V1,i+1(G)*di+1 +V2,i+1*Ri+1 + V2,i+1(G)*di+1-
V3,i+1* di+1 > Lk
=> 5.004*10 + 5.282*15 + 5.282*15 +5.282*10.385 +2.502*15 + 2.502*10.385 –
6.116*10.385 = 263.35 m > 250m, so it can be written as below:
V1,i *(Ri –Ei)+ V1,i(G) * Gi + V1,i+1*Ri+1 + V1,i+1(G)*ti+1 +V2,i+1*Ri+1 + V2,i+1(G)*ti+1 - V3,i+1* ti+1 = Lk or
Bus Arrival Time Estimation Using Traffic Signal Database & Shockwave Theory Md. Matiur Rahman
1, Lina Kattan
2, S.C. Wirasinghe
3 and Doug Morgan
4
12th WCTR, July 11-15, 2010 – Lisbon, Portugal
15
sec38.2116.6502.2282.5
15*502.215*282.515*282.510*004.5250
***)(*
1,3)(1,2)(1.1
11,211,1)(,1,11
iGiGi
iiiiiGiiiiki
VVV
RVRVGVERVLt
Step 4: Determination of link travel time
Cruising Travel time: Tc,k = (Ri –Ei) +Gi + Ri+1 + ti+1 = 10 + 15 + 15 + 2.38 = 42.38 sec
Travel time in queue:
82.46
38.2*116.638.2*502.215*502.21,
ikN
sec96.10
44.0
82.4
)(
1,,
tq
NT
s
ikkd
Link travel time, Tk = Tc,k + Td,k = 42.38 +10.96 = 53.34 sec
The aforementioned examples show that this approach can incorporate the intersection
delay well in real-time bus arrival time estimation for short distance between bus stop and
downstream signal as well as for the longer distance between bus stop and downstream
signal.
8. CONCLUSIONS
Delays at signalized intersection are critical factors in the estimation of real-time bus arrival
time. In this paper, a framework for incorporating this delay in the estimation of real-time bus
travel time was proposed. This framework divides the estimation of link travel time into two
components: 1) link cruising travel time estimation and 2) travel time needed to exit the
queue at the intersection. Exact signal and traffic information from the signal database and
surveillance system are assumed available online and integrated in this framework. The
proposed framework is based on the analysis of all possible cases that the bus might
encounter when travelling the distance between the bus stop and downstream signal. The
approach is tested on two numerical examples. It is expected that bus delays due to
presence of queues in signalized intersection can be addressed with reasonable accuracy in
this model.
Future work in this research will expand on the case of actuated rather than just pre-timed
signals. Future work will also incorporate Bus travel time on an arterial that considers the
presence of signal progression which might reduce the effect of shockwave propagation. In
addition, the framework could possibly be integrated with the online information on traffic
condition provided by the MIST (Management Information Systems for Transportation)
system available in more than half of the signalized intersections in Calgary.
ACKNOWLEDGEMENT
This research was supported in part by the PUTRUM Research Program funded by Calgary
Transit, and by NSERC. We wish to acknowledge the support given by Fred Wong, Director
of Calgary Transit.
Bus Arrival Time Estimation Using Traffic Signal Database & Shockwave Theory Md. Matiur Rahman
1, Lina Kattan
2, S.C. Wirasinghe
3 and Doug Morgan
4
12th WCTR, July 11-15, 2010 – Lisbon, Portugal
16
REFERENCES
1. Chen, M., Liu, X. Xia, J. and Chien, S.I. (2004). A Dynamic Bus-Arrival Time Prediction
Model Based on APC Data. Computer-Aided Civil and Infrastructure Engineering 19,
pp.364–376.
2. Chien, S.I., Ding, Y. and Wei, C. (2002). Dynamic Bus Arrival Time Prediction with
Artificial Neural Networks. Journal of Transportation Engineering, Vol.128, Issue 5, pp.
429-438.
3. Chowdhury, M. A. and Sadek, A.W. (2003). Fundamentals of Intelligent Transportation
Systems Planning. Artech House, pp-14-18.
4. Dailey, D.J., Maclean, S.D., Cathey, F.W. and Wall, Z.R. (2001). Transit Vehicle Arrival
Prediction Algorithm and Large Scale Implementation. Transportation Research Record:
Journal of the Transportation Research Board, No. 1771, Transportation Research Board
of the National Academies, Washington, D.C., pp. 46-51.
5. Dion, F., Rakha, H. and Kang, Y. (2004). Comparison of Delay Estimates at Under-
Saturated and Over-Saturated Pre-Timed Signalized Intersections. Transportation
Research Part B 38, pp. 99–122.
6. Farhan, A. (2002). Bus Arrival Time Prediction For Dynamic Operation Controls and
Passenger Information System. MSc thesis, Department of Civil Engineering, University
of Toronto.
7. Garber, N.J. and Hoel, L.A. Traffic and Highway Engineering. Cengage Learning, SI
Edition.
8. Geroliminis, N. and Skabardonis, A. (2005). Prediction of Arrival Profiles and Queue
Lengths along Signalized Arterials by Using a Markov Decision Process. Transportation
Research Record: Journal of the Transportation Research Board, No. 1934,
Transportation Research Board of the National Academies, Washington, D.C., pp. 116-
124.
9. Lighthill, M.J. and Whitham, G.B. (1955). On kinematic waves I Flood movement in long
rivers. II A theory of traffic flow on long crowded roads. Proc. Roy. Soc. (London) A229,
pp. 281–345.
10. Lin, W.H. and Zeng, J. (1999). Experimental Study of Real-Time Bus Arrival Time
Prediction with GPS Data. Transportation Research Record: Journal of the
Transportation Research Board, No. 1666, Transportation Research Board of the
National Academies, Washington, D.C., pp. 101-109.
11. Liu, H. X., Wu, X., Ma, W., and Hu, H. (2009). Real-Time Queue Length Estimation for
Congested Signalized Intersections. Transportation Research Part C, Volume 17, Issue
4, pp.412-427
12. Liu, H., van Zuylen, H., van Lint, H. and Salomons, M. (2006). Predicting Urban Arterial
Travel Time With State-Space Neural Networks And Kalman Filters. Transportation
Research Record: Journal of the Transportation Research Board, No. 1968,
Transportation Research Board of the National Academies, Washington, D.C., pp. 99-
108.
13. Patnaik, J., Chien, S. and Bladikas, A. (2004). Estimation of Bus Arrival Times Using
APC Data. Journal of Public Transportation, Vol. 7, No. 1, pp. 1-20.
Bus Arrival Time Estimation Using Traffic Signal Database & Shockwave Theory Md. Matiur Rahman
1, Lina Kattan
2, S.C. Wirasinghe
3 and Doug Morgan
4
12th WCTR, July 11-15, 2010 – Lisbon, Portugal
17
14. Son, B., Kim, H.J., Shin, C.H., and Lee, S.K. (2004). Bus Arrival Time Prediction Method
for ITS Application. KES 2004, LNAI 3215, Springer-Verlag Berlin Heidelberg, pp. 88–94.
15. Sun, D., Luo, H., Fu, L., Liu, W., Liao, X., Zhao, M. (2007). Predicting Bus Arrival Time on
The Basis of Global Positioning System Data. Transportation Research Record: Journal
of the Transportation Research Board, No. 2034, Transportation Research Board of the
National Academies, Washington, D.C., pp. 62-72.
16. van Hinsbergen, C.P.IJ. and van Lint, J.W.C. (2008). Bayesian Combination of Travel
Time Prediction Model. Transportation Research Record: Journal of the Transportation
Research Board, No. 2064, Transportation Research Board of the National Academies,
Washington, D.C., pp. 73-80.
17. Viti F. and Van Zuylen H.J. (2004). Modeling Queues At Signalized Intersections.
Proceedings of the 83th Annual Meeting of the Transportation Research Board,
Washington D.C., USA, National Academies Press.
18. Wall, Z. and Dailey, D.J. (2004). An Algorithm for Predicting the Arrival Time of Mass
Transit Vehicles Using Automatic Vehicle Location Data. Transportation Research Board,
78th Annual Meeting, Washington D.C.
19. Wirasinghe, S.C. (1978). Determination of Traffic Delays from Shock-wave Analysis.
Transportation Research, Volume 12, Issue 5, pp. 343-348.